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NBSIR 74-549^0
Nucleus-Nucleus Collisions at
Very High Energies
A. Bia/as
Max-Planck Institute for Physics and Astrophysics
Munchen, Germany
W. Czyz
Institute of Nuclear Physics
Cracow, Poland
L. C. Maximon
Center for Radiation Research
Institute for Basic Standards
National Bureau of Standards
Washington, D. C. 20234
August 1974
Work supported by:
NBS Special Foreign Currency Program, Grant No. (G) 173.
U. S. DEPARTMENT OF COMMERCE
NATIONAL BUREAU OF STANDARDS
NBSIR 74-549
NUCLEUS-NUCLEUS COLLISIONS AT
VERY HIGH ENERGIES
A. Bia/as
Max-Planck Institute for Physics and Astrophysics
Munchen, Germany
W. Czyz
Institute of Nuclear Physics
Cracow, Poland
L. C. Maximon
Center for Radiation Research
Institute for Basic Standards
National Bureau of Standards
Washington, D. C. 20234
August 1974
Work supported by:
NBS Special Foreign Currency Program, Grant No. (G) 173.
U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary
NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director
nucleus-nucleus collisions at very high energies
Ao Bialas
Max-Planck Institute for Physics and Astrophysics, Mttnchen
W. Czyz
Institute of Nuclear Physics, Cracow
L.C, Maximon
National Bureau of Standards, Washington D.C.
I. Introduction
It has been stressed many times that nucleus-nucleus
collisions at high energies are worth investigating "both
theoretically and experimentally [l-6^] because virtually
nothing is known about this exotic field and, at the same
time, the theoretical models one may envisage give several
simple and definite predictions. The present note discusses
one such model which was originally proposed for hadron-nucleus
scattering f 7] but can be straightforwardly generalized to
nucleus-nucleus collisions. The main purpose of this note is
to spell out this generalization and to give the formulae for
the total cross sections, diffractive production cross sections,
and inclusive single particle distributions in multiparticle
production in nucleus-nucleus collisions. One of the important
conclusions of this analysis is that the measurements of the
relations between various total cross sections proposed in
refs, £3-6] could test the model of ref . £7] •
^"work sponsored by NBS Special Foreign Currency Program, Grant No.(G)173.
*)On leave of absence from Institute of Physics, Jagellonian Univ.,Cracow, Poland, and Institute of Nuclear Physics, Cracow, Poland.
- 2
2 * The__model •
We shall work in the cm, system of the two nuclei
containing A and B nucleons. The S matrix for such ^collision
is, according to the model of ref.[~7J »
the
where j:* is the S matrix for /j-th nucleon from A colliding
with jf-th nucleon from B. As in 7 » the "aosorptive" part,A
Pjgj which describes both elastic scattering and diffractive
production is generated by the driving term T, responsible^
/ ~ t \for non-diffractive multiple production processes ( ~ 'j^ /
and the unitarity of V. givesJ*
h '
JS V- ' V •
C2>2;
Again, iollowing the approach of ref, [?]» we accept that
the production processes ^in fact all possible processes^ are
completely coherent over the two nuclei and this fact is
incorporated into our description in a relativistically
invariant manner by making the operators of Eqs. (2.l) and 0^*^)
dependent only on the transverse degrees of freedom . In the
present simple version we shall use only the impact parameter
- 2 -
and the transverse distances of the individual nucleons from
the axis of cylindrical symmetry of the collision as these
transverse degrees of freedom.
One should also keep in mind that, henceforth, we shall
deal only with sum rules in which the momentum transfer
between the two nuclei ( more precisely: between the two groups
of particles which emerge from the collision and go to the
left and to the right in the cm. system) is integrated over
and the sum performed over all possible excitations (of nuclei
and of their components: nucleons) • Gross sections for
specific processes are obtained, in the present approach,
through the introduction of sutable projection operators into
such sums and integrations. Note that in this approach one
completely avoids dealing with the longitudinal momentum
transfers in the process of multiple collisions: for example,
as can be seen in Sec. Con inclusive cross sections, the
dependence on the longitudinal momenta comes only through
the final expression, (2.6) , where they are accounted for
in the "elementary" hadron-nucleon inclusive cross section,
45;(W
•
Assuming the same properties of the individual
matrices as in ref. [7 J and going through algebra analogous
to that used in the case of hadron-nucleus collisions [7 J ,
we readily obtain:
A. Elastic scattering and coherent diffractive production .
The amplitude for both processes at the impact parameter
- 4 -
b reads (to lowest order in diffractive production)
A Ale,
>
(2.3)
0 £ (A)j <Sf
In this formula the absorptive part of the matrix was
split into the elastic and diffractive inelastic parts
where y.'n is diagonal ( this is the "profile" of the Glauber
\J
model! and the off-diagonal correction responsible for
diffractive production is assumed to be small. In ref . £7]
such a form for V. was explicitely obtained from theft
bremsstrahlung model of multi-particle production [bj .
It was also suggested by the general considerations ox ref.[9J
In Eq. (2,5) the states are labelled by the nuclear wave
functions of both nuclei Cp^ j (ft
^ y an<3 ^y the produced
- 5 -
particles, schematically denoted fh) • The first part of the
r.h.s. of (2.3) gives the well known nucleus-nucleus high
r -i theenergy elastic scattering amplitude [1J • Note that/second
part of the r.h.s. of (2.3] , which gives diffractive
production, is proportional to AB^ which means that the cross
2 2sections have a A B dependence. Hence it would seem that
diffractive production in nucleus-nucleus collisions may
contribute more significantly to, e.g., one particle inclusive
distributions (see below) than in the case of hadron-nucleus
collisions, where B = lo
B. The total cross sections . Similarly, as in [7] , after
employing closure 2 E K)$^fXtf ty**] ^ I
=I
and integrating over the momentum transfer between the two
nuclei, we obtain the following formula for the total nucleus-
nucleus cross section:
(2.4)
\where we employed unitarity: 5^ n S^g — I • Again, as in
£7] * the total cross section is, to zeroth order in <£ ,
the same as that given by the Glauber model refs* ("l,5>4-I ,
and all the relations between the total cross sections
proposed in refs. [3,4,5 >6jf » when compared with experiment,
will provide a test for the multiparticle production model
proposed in re£ [ 7J and used here.
Go Inclusive single particle distribution in multiparticle
production in nucleus-nucleus collisions . We again follow
the same algebra as in £7 J :
(2.5)
The commutation relations C a ^p; ); ^jj were worked
out in for the case of the hadronic bremsstrahlung model
When we employ them, we again obtain the result that the
"diagonal" terms give the leading contribution:
(2.6)
- 7 -
and the non-diagonal contributions appear to be small.
The same qualifications made regarding Eq. (ll) of
ref.\J?\ > viz., that corrections are to be expected in the
very forward direction, can be made here. However, we would
like to emphasize the following point. In performing the
algebraic operations which lead to (2.6^ or Eq. ^ll) of
ref. £7] one neglects energy-momentum conservation in Ji\,
which results in suppressing diffractive production . Hence
neither Eq. (2.6j nor Eq. (ll) of ref. contain
contributions to coming from diffractive production*,
In the case of hadron-nucleus interactions this simplification
may still be not too bad, but, as has already been pointed
out in this note, in the case of nucleus-nucleus collisions
it is more serious. It would seem that one could estimate
the relevant corrections starting directly from Eq. (2.3,) •
In any case one should expect the relation (2 .6] to fail in
the extreme forward and backward directions, which are
populated by products of diffractive processes.
In concluding, one may say that one can literally take
over the model of ref. £7] an<3 extend it trivially to thethe
case of nucleus-nucleus scattering gust as/ Glauber model for
hadron-nucleus elastic scattering can be extended to
nucleus-nucleus elastic scattering.
- 8 -
References
1. W. Czyz and L.C. Maximon, Ann.Phys. 52, 59 (JL969) §
Z. H.H. Heckman, High Energy Heavy Ions: A New Area for
Physics Research, 5th Int. Conf • on High-Energy Physics
and Nuclear Structure, Uppsala, June 1973*
3. P.M. Pishbane and J.S. Trefil, Phys. Rev. Letters 32, 396
(.1974),
4 0 V. Franco, Phys. Revo Letters ^2, 911 C-WM
5. S. Barshay, G.B. Dover and J. P. Vary, The Validity of the
Factorization Hypothesis for Nucleus-Nucleus Cross Sections
at High Energies, Brookhaven preprint BNL 15811, April, 1974.
6. S. Barshay, G.B. Dover and J. P. Vary, Nucleus-Nucleus
Cross Sections and the Validity of the Factorization
Hypothesis at Intermediate and High Energies, Brookhaven
preprint BNL 18972, May, 1974.
7. A. Bialas and W. Czyz, Short-time behavior of hadronic
interactions and the hadron-nucleus production processes,
Max-Planck Institute for Physics and Astrophysics preprint,
to be published in Physics Letters.
do A. Bialas and A. Kotahski, Acta Physica Polonica B4, 659
(1973) o
9. W. Czyz, Phys. Rev. D8, 3219 (1973) *
NBS-114A IHF. V. 7-73)
U.S. D€PT. OF" COMM.BIBLIOGRAPHIC DATA
SHEET
1. PUBLICATION OR REPORT NO.
NBSIR 74-549
2. Gov't AccessionNo.
3. Recipient's Acccsr.iun No.
4. TITLE AND SUBTITLE
Nucleus-Nucleus Collisions at Very High Energies
5. Publication Dace
August 1974
6. Performing Organization Code
7. AUTHOR(S)
A. Bialas, W. Czyz , and L. C. Maximon8. Performing Organ. Report No.
NBSIR 74-5499. PERFORMING ORGANIZATION NAME AND ADDRESS
NATIONAL BUREAU OF STANDARDSDEPARTMENT OF COMMERCEWASHINGTON, D.C. 20234
10. Project/Task/Work Unit No.
2400103
11. Contract/Grant No.
12. Sponsoring Organization Name and Complece Address (Street, City, State, ZIP)
Same as No. 9
13. Type of Report & PeriodCovered
14. Sponsoring Agency Code
15. SUPPLEMENTARY NOTES
16. ABSTRACT (A 200-word or less factual summary ol most significant information. If document includes a significant
bibliography or literature survey, mention it here.)
A model for high energy nucleus-nucleus collisions is presented. Formulae aregiven for the total cross sections, diffractive productive cross sections, andinclusive- single particle distributions in multiparticle production in nucleus-nucleus collisions. It is noted that measurement of the relations between a fewnucleus-nucleus total cross sections could be used to test this model.
17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a propername; separated by semicolons)
Diffractive production cross sections; glauber model; high eneVgy hadron interactions;inclusive single particle distributions; nucleus-nucleus collisions; total hadron
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